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Properties

Label 8-1620e4-1.1-c3e4-0-5
Degree $8$
Conductor $6.887\times 10^{12}$
Sign $1$
Analytic cond. $8.34688\times 10^{7}$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·5-s − 13·7-s − 15·11-s − 10·13-s + 90·17-s + 86·19-s − 105·23-s + 25·25-s − 105·29-s + 71·31-s + 130·35-s − 298·37-s − 390·41-s − 169·43-s − 345·47-s + 636·49-s + 1.50e3·53-s + 150·55-s − 900·59-s − 61-s + 100·65-s + 335·67-s + 3.30e3·71-s + 314·73-s + 195·77-s − 346·79-s − 600·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.701·7-s − 0.411·11-s − 0.213·13-s + 1.28·17-s + 1.03·19-s − 0.951·23-s + 1/5·25-s − 0.672·29-s + 0.411·31-s + 0.627·35-s − 1.32·37-s − 1.48·41-s − 0.599·43-s − 1.07·47-s + 1.85·49-s + 3.88·53-s + 0.367·55-s − 1.98·59-s − 0.00209·61-s + 0.190·65-s + 0.610·67-s + 5.51·71-s + 0.503·73-s + 0.288·77-s − 0.492·79-s − 0.793·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(8.34688\times 10^{7}\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.783372610\)
\(L(\frac12)\) \(\approx\) \(4.783372610\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
good7$D_4\times C_2$ \( 1 + 13 T - 467 T^{2} - 650 T^{3} + 228880 T^{4} - 650 p^{3} T^{5} - 467 p^{6} T^{6} + 13 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 15 T - 17 p T^{2} - 33750 T^{3} - 1901292 T^{4} - 33750 p^{3} T^{5} - 17 p^{7} T^{6} + 15 p^{9} T^{7} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 + 10 T - 2843 T^{2} - 14510 T^{3} + 3614740 T^{4} - 14510 p^{3} T^{5} - 2843 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 - 45 T + 8026 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 43 T - 1410 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 105 T - 13759 T^{2} + 47250 T^{3} + 332069592 T^{4} + 47250 p^{3} T^{5} - 13759 p^{6} T^{6} + 105 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 105 T - 38203 T^{2} + 47250 T^{3} + 1559683938 T^{4} + 47250 p^{3} T^{5} - 38203 p^{6} T^{6} + 105 p^{9} T^{7} + p^{12} T^{8} \)
31$D_4\times C_2$ \( 1 - 71 T - 55709 T^{2} - 82928 T^{3} + 2652882388 T^{4} - 82928 p^{3} T^{5} - 55709 p^{6} T^{6} - 71 p^{9} T^{7} + p^{12} T^{8} \)
37$D_{4}$ \( ( 1 + 149 T + 86100 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 390 T + 59258 T^{2} - 17550000 T^{3} - 6613351377 T^{4} - 17550000 p^{3} T^{5} + 59258 p^{6} T^{6} + 390 p^{9} T^{7} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 + 169 T - 133073 T^{2} + 442780 T^{3} + 17533387480 T^{4} + 442780 p^{3} T^{5} - 133073 p^{6} T^{6} + 169 p^{9} T^{7} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 + 345 T - 60721 T^{2} - 9625500 T^{3} + 9171876612 T^{4} - 9625500 p^{3} T^{5} - 60721 p^{6} T^{6} + 345 p^{9} T^{7} + p^{12} T^{8} \)
53$D_{4}$ \( ( 1 - 750 T + 429154 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 900 T + 233642 T^{2} + 149040000 T^{3} + 123651020523 T^{4} + 149040000 p^{3} T^{5} + 233642 p^{6} T^{6} + 900 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + T - 214019 T^{2} - 239942 T^{3} - 5716040942 T^{4} - 239942 p^{3} T^{5} - 214019 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 5 p T - 151217 T^{2} + 1690420 p T^{3} - 54809822480 T^{4} + 1690420 p^{4} T^{5} - 151217 p^{6} T^{6} - 5 p^{10} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 1650 T + 1313422 T^{2} - 1650 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 157 T + 763440 T^{2} - 157 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 346 T - 564191 T^{2} - 104551166 T^{3} + 165616672204 T^{4} - 104551166 p^{3} T^{5} - 564191 p^{6} T^{6} + 346 p^{9} T^{7} + p^{12} T^{8} \)
83$D_4\times C_2$ \( 1 + 600 T + 48926 T^{2} - 499500000 T^{3} - 436016659893 T^{4} - 499500000 p^{3} T^{5} + 48926 p^{6} T^{6} + 600 p^{9} T^{7} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 - 1500 T + 1640338 T^{2} - 1500 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 623 T - 1456667 T^{2} - 12117350 T^{3} + 2216065417870 T^{4} - 12117350 p^{3} T^{5} - 1456667 p^{6} T^{6} - 623 p^{9} T^{7} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.47715581278334605242972451699, −6.09110405117059757715037826988, −5.73007749789249656324601020149, −5.67104664148513234714497972993, −5.52328543625860939335970644463, −5.16668772822770495107353162723, −4.89899233156002580901529838420, −4.89618407160046825574719332960, −4.83099373519599163331704781349, −4.00954302176275887816087713611, −3.95352472336723462470569207640, −3.87375774338204350875735033111, −3.76083967485491057766468119154, −3.26377679197488960829707892714, −3.15733138707328541727610129701, −3.09957054149778554872326080061, −2.68487171441241336226363608888, −2.15187702896376035357717661801, −1.99559088519357962679497357648, −1.99170877021759190329343580410, −1.52973220953015020772436594560, −0.881368529721790449658195195640, −0.71081366311811625446271859026, −0.60644622250810316576767405203, −0.31620939483641639636445961774, 0.31620939483641639636445961774, 0.60644622250810316576767405203, 0.71081366311811625446271859026, 0.881368529721790449658195195640, 1.52973220953015020772436594560, 1.99170877021759190329343580410, 1.99559088519357962679497357648, 2.15187702896376035357717661801, 2.68487171441241336226363608888, 3.09957054149778554872326080061, 3.15733138707328541727610129701, 3.26377679197488960829707892714, 3.76083967485491057766468119154, 3.87375774338204350875735033111, 3.95352472336723462470569207640, 4.00954302176275887816087713611, 4.83099373519599163331704781349, 4.89618407160046825574719332960, 4.89899233156002580901529838420, 5.16668772822770495107353162723, 5.52328543625860939335970644463, 5.67104664148513234714497972993, 5.73007749789249656324601020149, 6.09110405117059757715037826988, 6.47715581278334605242972451699

Graph of the $Z$-function along the critical line